Introduction to Aircraft Aeroelasticity and Loads

2

Vibration of Multiple Degree of Freedom Systems

In this chapter, some of the basic concepts of vibration analysis for multiple degree of freedom (MDoF) discrete parameter systems will be introduced, as there are some significant differences to a single degree of freedom (SDoF) system. The term ‘discrete (or sometimes lumped) parameter’ implies that the system in question is a combination of discrete rigid masses (or components) interconnected by flexible stiffness and damping elements. Note that the same approaches may be employed when a modal coordinate system is used (see later). On the other hand, ‘continuous’ systems, considered later in Chapters 3 and 4, are those where all components of the system are flexible/elastic and deform in some manner.

The focus of this chapter will be in setting up the equations of motion, finding natural frequencies and mode shapes for free vibration and determining the forced vibration response with various forms of excitation relevant to aircraft loads. Some of the core solution methods introduced in Chapter 1 will be considered for MDoF systems. For simplicity, the ideas will be illustrated for only two degrees of freedom. The general form of equations will be shown in matrix form to cover any number of degrees of freedom, since matrix algebra unifies all MDoF systems. Further treatment may be found in Tse et al. (1978), Newland (1989), Rao (1995), Thomson (1997) and Inman (2006).

2.1 SETTING UP EQUATIONS OF MOTION

There are a number of ways ...